Course notes for EE394V Restructured Electricity Markets: Market Power

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1 Course notes for EE394V Restructured Electricity Markets: Market Power Ross Baldick Copyright c 2009 Ross Baldick Title Page 1 of 54 Go Back Full Screen Close Quit

2 1 Background This review of background material is based, in part, on: Part 1 of Power System Economics, by Steven Stoft, and EE394V, Locational Marginal Pricing. Title Page 2 of 54 Go Back Full Screen Close Quit

3 Outline (i) Restructured electricity markets, (ii) Variable operating costs, (iii) Marginal costs, (iv) Price-taking assumption, (v) Offer-based economic dispatch, (vi) Example, (vii) Fixed demand and pivotal offers, (viii) Computational issues, (ix) Forward contracts, (x) Summary, (xi) Homework exercises: Due Tuesday, February 9 by 10pm, via , and each Tuesday thereafter. Title Page 3 of 54 Go Back Full Screen Close Quit

4 1.1 Restructured electricity markets Until the 1990s, most electricity in North America was supplied by regulated utilities that each had a franchise service area: utility had right to sell electricity at retail in franchise service area, prices for delivered energy to retail customers set by state body and were chosen to allow an opportunity to earn a mandated level of profit, some wholesale trade between utilities and other entities, but competition for sales to retail customers essentially prohibited, cost-of-service regulation of monopoly. Rationale for such arrangements was that a single firm could provide services more cheaply than multiple competing firms because of economies of scale: cost of one firm providing all services lower than sum of costs of multiple firms providing services. An industry where it is cheapest for a single firm to provide all services is called a natural monopoly. Title Page 4 of 54 Go Back Full Screen Close Quit

5 Restructured electricity markets, continued Typical criticisms of such arrangements include lack of incentives to innovate and lack of incentives to keep total capital and operating costs at minimum. Large-scale transmission networks, among other things, have allowed for a different arrangement where multiple owners of generators compete to sell their energy in a restructured electric market: geographical scope of restructured market, (for example ERCOT,) covers what were the franchise service areas of multiple regulated utilities (TXU, Houston Lighting and Power, etc), transmission and distribution functions remain regulated under assumption that building and operating transmission and distribution are still natural monopolies, typically consolidating the transmission operations of multiple utilities under an Independent System Operator (ISO, for example ERCOT), fierce competition amongst generation firms provides stronger incentives to minimize costs and (arguably) incentives for innovation, but, not clear that competition amongst firms is fierce! Title Page 5 of 54 Go Back Full Screen Close Quit

6 Restructured electricity markets, continued Purpose of this course is to understand competitive interactions between generation firms in restructured electricity markets: interaction between several, but not a huge number of, firms that are assumed to maximize their profits over the choices they can make, emphasize unique aspects of electricity, such as transmission and the need to balance supply and demand collectively, which significantly affect operation of electricity markets. For reasons that will become clearer, we will usually neglect issues related to limited competition between purchasers of energy. Title Page 6 of 54 Go Back Full Screen Close Quit

7 1.2 Variable operating costs To produce electricity, generators incur operating costs, which include fuel and other costs, such as maintenance: these costs would be avoided if the generator were out of service. To emphasize the distinction between: those operating costs that depend on the level of production, and fixed costs such as construction costs, fixed maintenance costs, and operating costs such as start-up costs that are not directly related to production level, we will say variable operating costs. To emphasize the distinction between: the total variable operating costs of producing at a particular level, the derivative of the variable operating costs, and the average of the variable operating costs (variable operating costs divided by production), we sometimes say total variable operating costs for variable operating costs. Phrases used to describe these concepts are not completely uniform. Title Page 7 of 54 Go Back Full Screen Close Quit

8 Variable operating costs, continued Assume each generator i has total variable operating costs c i :R R that specify the cost of operating (in dollars per hour) versus generation level. Recall that c i :R R is shorthand for saying that c i is a function that takes a real-valued argument (specified by the setr) and returns a real value (also specified by the setr). In particular, c i (Q i ) is the cost per hour of operating at the production level Q i. We will follow the economics convention of writing q or Q for quantity and p or P for price (not reactive and real power we will stay away from reactive power prices!) We will also follow the economics convention of graphing prices on the vertical axis and quantity on the horizontal axis, whichever is the independent variable in a particular formulation. c i may only be useful in some operating range, such as [q i,q i ]. Recall that the notation [q i,q i ] means the closed interval between some lower capacity limit q i and some upper capacity limit q i. Title Page 8 of 54 Go Back Full Screen Close Quit

9 Variable operating costs, continued We typically assume that this variable operating cost function is convex or bowl-shaped on [q i,q i ], although this often simplifies reality. c i (Q i ) Q i Fig Example of convex total variable operating costs c i (Q i ) versus production Q i for a generator. q i q i Title Page 9 of 54 Go Back Full Screen Close Quit

10 Variable operating costs, continued Convex functions have a number of desirable properties. For example, convex functions are differentiable almost everywhere; that is, except at a finite number or countably infinite number of points. $165,000/h $125,000/h c i (Q i ) $50,000/h Q i (MW) Fig Second example of convex total variable operating costs c i (Q i ) versus production Q i for a generator. Title Page 10 of 54 Go Back Full Screen Close Quit

11 Variable operating costs, continued Typically, convex functions are piece-wise continuously differentiable. The derivative of the total variable costs function from Figure 1.2 is well-defined over the open interval (0,4500) except at Q i = 2500,4000. c i (Q Q i ) i $80/MWh $50/MWh $20/MWh Q i (MW) Fig Derivative of convex variable operating costs from Figure 1.2. Title Page 11 of 54 Go Back Full Screen Close Quit

12 1.3 Marginal costs The derivative of total variable operating costs is called the marginal costs, in dollars per megawatt-hour. At the points of discontinuity of the marginal costs, the value jumps up from the left-hand marginal cost to the right-hand marginal cost. We also define the marginal cost range to be the interval between the left-hand and right-hand marginal cost. A property of a convex function is that its derivative is a non-decreasing function: marginal costs are non-decreasing, as shown in Figure 1.3. Title Page 12 of 54 Go Back Full Screen Close Quit

13 Marginal costs, continued At each point of differentiability of total variable operating costs: the left-hand and right-hand marginal costs are the same, the marginal costs are continuous, and the marginal cost range is a single point, namely equal to the derivative. The left-hand marginal cost, evaluated at production Q i, shows the savings from reducing production below Q i. The right-hand marginal cost, evaluated at production Q i, shows the extra cost of increasing production above Q i. At maximum production q i, we can think of the right-hand marginal cost as being infinite: as suggested by the upward arrow at quantity 4500 MW in the marginal costs in Figure 1.3. At minimum production q i, we can think of the left-hand marginal cost as being negative infinite: as suggested by the downward arrow at quantity 0 MW in the marginal costs in Figure 1.3. Title Page 13 of 54 Go Back Full Screen Close Quit

14 Marginal costs, continued In essentially all cases, from a practical perspective, we can imagine smoothing out the jumps in the marginal cost curve by assuming a thin interval of production over which marginal cost raises rapidly: Over a few megawatts, say, the marginal cost increases from the left-hand to the right-hand marginal cost. At maximum production, the marginal cost rises from the left-hand marginal cost towards infinity. Results with the smoothed marginal cost curve will almost always be essentially the same as with the discontinuous marginal cost curve and are often easier to interpret. Title Page 14 of 54 Go Back Full Screen Close Quit

15 1.4 Price-taking or competitive behavior A generator is a price-taker in the economics sense or behaves competitively if, given a price, it sets its production (or, arranges that its production is set) so that the marginal cost range contains the price: this use of the phrase price-taker in an economics sense to behave competitively is different to a typical use of this phrase in electricity markets where, for example, a generator schedules its production and takes whatever the price happens to be. We will see that a price-taker in the electricity markets sense may not be behaving competitively! We will be careful to distinguish these two uses of the phrase price-taker! Title Page 15 of 54 Go Back Full Screen Close Quit

16 Price-taking behavior, continued If the marginal costs are continuous then a price-taker in the economics sense (that is, a generator behaving competitively) chooses production so that marginal cost equals price. With the smoothed-out marginal cost curve interpretation, a price-taker in the economics sense always chooses production so that marginal cost equals price: However, note that at the maximum production level, the smoothed marginal cost rises above the left-hand marginal cost. With piece-wise continuously differentiable marginal costs, a price-taker in the economics sense always chooses production so that the marginal cost range contains the price. Title Page 16 of 54 Go Back Full Screen Close Quit

17 Price-taking behavior, continued An analogous definition applies to load: the focus is on the benefits of consumption as a function of demand b k (D k ) and on the derivative of benefits, which is called the willingness-to-pay. A price taking load chooses demand such that willingness-to-pay equals price (or willingness-to-pay range contains the price). In the context of benefits of consumption, the notion of a fixed demand (that is, not associated with any finite willingness-to-pay) is meaningless since it implies an infinite desire for consumption! We will usually think of demands as having either: an explicit willingness-to-pay as specified by its bid, or an implicit proxy to the dis-benefit of involuntary curtailment, which is called the value of lost load (VOLL), and which is usually specified in practice by a price or offer cap. We will see that a fixed demand independent of price, or having a very high willingness-to-pay, is also problematic in the context of market power! Title Page 17 of 54 Go Back Full Screen Close Quit

18 1.5 Offer-based economic dispatch We assume an electricity market where owners of generators make offers to sell electricity and representatives of demand make bids to buy electricity to an independent system operator (ISO): an offer for generator i is a function p i :R R from quantity to price that specifies, for each quantity Q i produced, the minimum price p i (Q i ) to produce at that quantity; a bid for demand k is a function w k :R R from quantity to price that specifies, for each quantity D k consumed, the maximum willingness-to-pay w k (D k ) for that quantity, an offer or bid cap limits the highest allowable value of p i or w k : currently $2250/MWh in ERCOT, an offer or bid floor limits the lowest allowable value of p i or w k. Title Page 18 of 54 Go Back Full Screen Close Quit

19 Offer-based economic dispatch, continued The ISO seeks a price P to match supply and demand. That is, the ISO finds quantities Q i of production for each generator and quantities D k for each demand such that: total generation equals total demand: generators i Q i = demands k D k, for each generator i: Q i satisfies p i (Q i ) P p i (Q + i ), where Q+ i is any quantity larger than Q i, so that generator i is willing to produce Q i for price P, and would not prefer (based on its offer) to be producing more at this price than Q i, for each demand k: D k satisfies w k(d k ) P w k (D + k ), where D+ k is any quantity larger than D k, so that demand k is willing to pay P to consume D k, and would not prefer (based on its bid) to be consuming more at this price than D k. Title Page 19 of 54 Go Back Full Screen Close Quit

20 Offer-based economic dispatch, continued If we assume that the bids and the offers are continuous then these conditions simplify: total generation equals total demand: generators i Q i = demands k D k, for each generator i, Q i satisfies p i (Q i )=P, and for each demand k D k satisfies w k(d k )=P. The quantities Q i and D k and price P are calculated by the offer-based economic dispatch algorithm: other issues such as reserves can also be incorporated, resulting in prices and quantities for reserves etc. other issues such as start-up costs can also be incorporated, but this greatly complicates the economic dispatch problem to being a unit commitment problem, which is in general non-convex. We will focus on issues related to market power in energy and will not discuss in detail market power in ancillary services markets nor in the context of start-up costs. Title Page 20 of 54 Go Back Full Screen Close Quit

21 Offer-based economic dispatch, continued A price that matches supply and demand is called market clearing or clears the market: that is, supply equals demand, given the offers and bids. The process can be described as a type of auction: the ISO is an auctioneer, and many results from auction theory in economics can be used to help analyze electricity markets. Not all ISO rules result in market clearing prices: For example, the ISO may always use the highest offer price (the marginal offer price, ) even if demand is curtailed or if a higher demand bid willingness-to-pay is actually the market clearing price. ISO rules that deviate from market clearing prices are usually based on a desire to keep prices low. We will, however, usually assume that the ISO sets a market clearing price. Title Page 21 of 54 Go Back Full Screen Close Quit

22 Offer-based economic dispatch, continued Suppose that a generator is a price-taker in the economics sense (that is, behaves competitively) in a market where market clearing prices are used. What offer should it make into such a market? By definition, being a price-taker in the economics sense (that is, behaving competitively) means arranging for generation such that the marginal cost range contains the price. Title Page 22 of 54 Go Back Full Screen Close Quit

23 Offer-based economic dispatch, continued Suppose that, for each quantity Q i, the generator sets its offer p i (Q i ) equal to its marginal cost c i (Q Q i ) at Q i : i if there is a jump in the marginal cost at quantity Q i, then set p i (Q i ) equal to the left-hand marginal cost at Q i. Then, since the price P satisfies p i (Q i ) P p i (Q + i ) at the quantity Q i desired by the ISO, the generator will be acting a price-taker in the economics sense by generating at this level (that is, behaving competitively) since the marginal cost range will contain the price. We call this a price-taking or competitive offer. Similarly, a price-taking or competitive bid is where the bid equals the (left-hand) marginal benefit or willingness-to-pay of consumption: we will usually assume that demand is acting as a price-taker in the economics sense, with bid equal to the marginal benefit. Title Page 23 of 54 Go Back Full Screen Close Quit

24 Offer-based economic dispatch, continued What is special about price-taking (that is, competitive) behavior? Benefits of consumption minus costs of production is called the net surplus or sometimes the total surplus. With price-taking (that is, competitive) offers and bids, the matching of supply and demand corresponds to maximizing the net surplus. To see this, consider the problem of maximizing the net surplus: { max Q i,d k b k (D k ) c i (Q i ) } Q i = D k. Assuming the c i and b k are differentiable and ignoring capacity constraints, the first-order necessary conditions for optimizing this problem are that there is a price P such that: i,k, c i Q i (Q i) = b k D k (D k )=P, Q i = D k. That is, matching of supply and demand with competitive offers and bids results in maximizing net surplus. Title Page 24 of 54 Go Back Full Screen Close Quit

25 Offer-based economic dispatch, continued Similarly, with piece-wise continuously differentiable costs and benefits, maximizing net surplus involves a price P such that: the marginal cost range of each generator contains the price P, the marginal benefit range of each demand contains the price P, and supply and demand is matched. That is, with competitive offers, clearing the market results in maximizing the surplus. Offer-based economic dispatch can be seen as an auction where a price is sought that: clears the market, and maximizes surplus, given that offers reflect marginal costs and bids reflect marginal benefits. This price is called the competitive price and is where the competitive offers and bids intersect. Title Page 25 of 54 Go Back Full Screen Close Quit

26 Offer-based economic dispatch, continued The competitive price is often, but not always, equal to the highest accepted generation offer price, (which is called the marginal offer. ) Sometimes, the competitive price is equal to the willingness-to-pay of demand, or, in the case of curtailment, equals a proxy to the willingness-to-pay, such as value of lost load. Unless demand actively bids its willingness-to-pay, it may be difficult to determine this willingness-to-pay: difficult to determine the competitive price in the case of a given demand when generation capacity is limited. Traditional focus of electricity industry and, until recently, electricity markets, has been on meeting a given demand, assuming there is enough generation capacity available: problematic in context of market power (prices may be above competitive), and problematic in context of resource adequacy (prices may be below competitive and therefore not provide enough revenue to cover both operating costs and fixed costs). Title Page 26 of 54 Go Back Full Screen Close Quit

27 Offer-based economic dispatch, continued The competitive price and the corresponding supply quantity is called the competitive equilibrium. The basic pricing rule is that all demand pays and all generation is paid this price. Given the market clearing price P and competitive offers, the market clearing quantities maximize the profits ( producer surplus ) of the generators and maximize the consumer surplus of the loads: for each generator i, for the given price P, the choice Q i maximizes the producer surplus P Q i c i (Q i ) over choices of Q i, and for each demand k, for the given price P, the choice D k maximizes the consumer surplus b k (D k ) P D k over choices of D k. That is, the price aligns the profit-maximizing incentives of demand and generation to be consistent with maximizing the net surplus. Title Page 27 of 54 Go Back Full Screen Close Quit

28 Offer-based economic dispatch, continued When transmission constraints limit choices of generation, the prices will vary by bus, leading to locational marginal pricing. prices vary by bus, at any particular bus, all demand pays and all generation is paid the same price. Title Page 28 of 54 Go Back Full Screen Close Quit

29 Offer-based economic dispatch, continued Why would a generator behave as a price-taker in the economics sense and submit price-taking (that is, competitive) offers? If there are many competitors in the market then being a price-taker in the economics sense (that is, competitive) is profit maximizing! If there are few competitors then price-taking (competitive behavior) is not profit maximizing and the generator has market power. Market power is one reason why offer-based economic dispatch may fail to result in the competitive equilibrium: Other reasons may include errors in market design (such as if the pricing rule does not result in market clearing prices), Errors in market design may exacerbate market power. As mentioned, we will generally assume that the number of purchasers of energy is large enough so that they can be assumed to behave competitively: that is, demand bids will be assumed equal to derivative of benefits, but this may be violated when the ISO acts on behalf of demand collectively. Title Page 29 of 54 Go Back Full Screen Close Quit

30 1.6 Example Suppose that there are three types of generators: baseload, ten units each of capacity 250 MW with marginal cost of $20/MWh, total baseload capacity 2500 MW. intermediate, ten units each of capacity 150 MW with marginal cost of $50/MWh, total intermediate capacity 1500 MW. peaking, ten units each of capacity 50 MW with marginal cost of $80/MWh, total peaking capacity 500 MW. Total capacity is 4500 MW. Ignore start-up and min-load costs and variation of marginal cost with production for each type of generator. Assume that prices are chosen to clear the market and that the generators offer competitively. Title Page 30 of 54 Go Back Full Screen Close Quit

31 Example, continued Competitive offers from the generators mean offers with price: equal to marginal cost for quantities over the range from zero megawatts up to generation capacity, and infinite price for quantities higher than capacity (or price equal to the maximum price allowed by the market rules, the price cap, ) and negative infinite price for negative quantities. Adding up the 30 offers horizontally yields the competitive supply q c. The inverse of q c is the corresponding industry marginal cost function or competitive offer p c : marginal cost is $20/MWh for zero to 2500 MW, marginal cost is $50/MWh for 2500 MW to 4000 MW, marginal cost is $80/MWh for 4000 MW to 4500 MW, infinite for higher quantities, and negative infinite for negative quantities. Note the jumps in offer prices at 0 MW (from negative infinite offer price), 2500 MW, 4000 MW, and 4500 MW (to infinite offer price). Title Page 31 of 54 Go Back Full Screen Close Quit

32 Example, continued Price $80/MWh $50/MWh Competitive supply q c $20/MWh Quantity (MW) Fig Competitive supply for example. Title Page 32 of 54 Go Back Full Screen Close Quit

33 Example, continued Suppose that the demand bid was 2800 MW with a willingness-to-pay of $500/MWh. The price would be $50/MWh and all 2800 MW of demand would be served, with 2500 MW generated by baseload and 300 MW by intermediate. $80/MWh Price Demand bid $50/MWh Ú Competitive supply q c $20/MWh Quantity (MW) Fig Market clearing for 2800 MW demand for example. Title Page 33 of 54 Go Back Full Screen Close Quit

34 Example, continued Suppose that the demand bid was 3500 MW with a willingness-to-pay of $500/MWh. The price would be $50/MWh, all 3500 MW of demand would be served, with 2500 MW generated by baseload and 1000 MW by intermediate. Suppose that the demand bid was 4200 MW with a willingness-to-pay of $500/MWh. The price would be $80/MWh, all 4200 MW of demand would be served, with 2500 MW generated by baseload, 1500 MW by intermediate, and 200 MW by peaking. Title Page 34 of 54 Go Back Full Screen Close Quit

35 Example, continued Suppose that the demand bid was 4900 MW with a willingness-to-pay of $500/MWh. The price would be $500/MWh and only 4500 MW of demand would be served, with 2500 MW generated by baseload, 1500 MW by intermediate, and 500 MW by peaking. $800/MWh Price Competitive supply q c $500/MWh Demand bid Ú Quantity (MW) Fig Market clearing for 4900 MW demand for example. Title Page 35 of 54 Go Back Full Screen Close Quit

36 Example, continued How about demands of 2500 MW, 4000 MW, 4500 MW, which corresponds to jumps in offers? In these cases, the supply and demand intersect in a vertical segment. That is, there is a range of market clearing prices corresponding to the segment of overlap: 2500 MW Any price in the range $20/MWh to $50/MWh, 4000 MW Any price in the range $50/MWh to $80/MWh, 4500 MW Any price in the range $80/MWh to $500/MWh. In practice: can specify a tie-breaking rule such as lowest price in range, or computational implementation will pick a particular price. One tie-breaking rule involves a price based on a thought experiment: suppose that, in addition to actual generators, we are given a costless MW of generation, what is the value of that MW in terms of savings from reducing generation or increased benefits of serving more demand? set price equal to this value, called the marginal surplus. Title Page 36 of 54 Go Back Full Screen Close Quit

37 1.7 Fixed demand and pivotal offers Meeting fixed demand If we take the traditional view of meeting fixed demand, we face the problem that there may be insufficient demand to meet supply. Suppose that the demand was fixed at 4900 MW. $80/MWh Price Fixed Demand $50/MWh Competitive supply q c $20/MWh Quantity (MW) Fig Market does not clear with fixed 4900 MW demand. Title Page 37 of 54 Go Back Full Screen Close Quit

38 Meeting fixed demand, continued Taken literally, this situation results in involuntary curtailment of 400 MW: in practice, system operator may use deploy reserves to avoid curtailment. There is no price that equates supply and demand. There is no well-defined price in case of curtailment, unless we re-interpret fixed demand as really being a demand bid with a high willingness-to-pay, as in the previous example. Title Page 38 of 54 Go Back Full Screen Close Quit

39 1.7.2 Market power implications Note that all offers are needed to meet more than 4500 MW of demand. Highest priced offer will always be accepted in this case. Example assumed competitive offers, but firm owning a peaking unit could increase profits by increasing offer price. Actually, any firm could increase profits for a fixed demand of more than 4500 MW by offering above $80/MWh, since some production from every generator is required. A firm whose capacity is needed to meet fixed demand is called pivotal: in absence of market power mitigation, firm can offer at any price and be sure of being at least partially dispatched. Title Page 39 of 54 Go Back Full Screen Close Quit

40 Market power implications, continued Assuming each generator is owned by a different firm: for fixed demand more than 4450 MW, all firms are pivotal, for fixed demand more than 4350 MW up to 4450 MW, all except peaking units are pivotal, for fixed demand more than 4250 MW up to 4350 MW, baseload are pivotal, for fixed demand less than or equal to 4250 MW, no firms are pivotal. If firms own multiple generators, then firms can be pivotal for demands below 4250 MW. Being pivotal is an extreme form of market power. More subtle forms of market power can occur even if no firm is pivotal. Defining, understanding, modeling, and other aspects of market power are the topics of this course. Title Page 40 of 54 Go Back Full Screen Close Quit

41 1.8 Computational issues The dispatch and prices are solved as an optimization problem. The objective is the revealed benefits minus the revealed costs: revealed costs: c i (Q i ) = revealed benefits: b k (D k ) = Qi 0 Dk 0 p i (Q)dQ, w k (D)dD, where p i and w k is the corresponding offer and bid. The offered capacity and bid demand specifies upper and lower bound constraints: generation: 0 Q i Q i, demand: 0 D k D k, where Q i is the offered capacity and D k is the quantity where willingness-to-pay falls to zero. The power balance equality constraint is generators i Q i = demands k D k. Title Page 41 of 54 Go Back Full Screen Close Quit

42 Computational issues, continued (Revealed) surplus (or revealed benefits minus costs) is maximized subject to the upper and lower bound constraints and the power balance constraint: with price taking (that is, competitive) offers and bids: c k = c k, revealed costs equals actual costs, and b k = b k, revealed benefits equals actual benefits, so that the results of offer-based economic dispatch would maximize benefits minus costs. The Lagrange multiplier on the equality constraint in the solution is a market clearing price. Title Page 42 of 54 Go Back Full Screen Close Quit

43 1.9 Forward contracts Because prices will vary with supply and demand, market participants are exposed to the risk of high or low prices. It is possible to lock-in an agreed price for an agreed quantity in a forward contract. In an offer-based economic dispatch market, the most natural forward contract takes the form of a contract for differences. Title Page 43 of 54 Go Back Full Screen Close Quit

44 Forward contracts, continued For example, suppose that a generator and demand agree to a forward contract for 10 MW at $50/MWh: that is, the generator commits to providing 10 MW to demand at a net price paid by demand of $50/MWh. If the market clearing price is actually P then the demand will pay 10 MW Pto the ISO: To make the net payment by the demand equal to 10 MW $50/MWh, the demand should pay to the generator: 10 MW (50 $/MWh P). the forward contract is implemented as an agreement by the demand to pay this amount, called a contract for differences. General form of payment under contract for differences: (Contract quantity) ( (Contract price) P). There are many variations on this arrangement, but this description of contract for differences will provide a useful model of forward contracts. Title Page 44 of 54 Go Back Full Screen Close Quit

45 1.10 Summary (i) Restructured electricity markets, (ii) Variable operating costs, (iii) Marginal costs, (iv) Price-taking assumption, (v) Offer-based economic dispatch, (vi) Example, (vii) Fixed demand and pivotal offers, (viii) Computational issues, (ix) Forward contracts. Title Page 45 of 54 Go Back Full Screen Close Quit

46 Homework exercise: Due Tuesday, February 9, by 10pm Break into ten groups, groups g=1,2,3,4,5,...,10, of approximately three people each. Each group will be assigned a portfolio of three generators. The task for each group is to find offers that maximize the operating profit (revenue minus generation costs) for the group over a day s operation. Title Page 46 of 54 Go Back Full Screen Close Quit

47 Types of generators Suppose that types of generators are as in the previous example: baseload, capacity 250 MW with marginal cost of $20/MWh. intermediate, capacity 150 MW with marginal cost of $50/MWh. peaking, capacity 50 MW with marginal cost of $80/MWh. Ignore start-up and min-load costs and variation of marginal cost with production. Groups have different mixes of generation: 1, 2 three baseload units, 3, 4 three intermediate units, 5, 6 three peaking units, 7, 8, 9, 10 one of each type of unit. So there are ten units of each type in total, as in the previous example. Title Page 47 of 54 Go Back Full Screen Close Quit

48 Demand There are three pricing intervals t = 1,2,3, each of eight hours duration, with demand: (i) 2800 MW, (ii) 3500 MW, and (iii) 4200 MW, respectively, with willingness-to-pay of $500/MWh. Prices with competitive offers From the previous example, we know that if the generators were offered competitively at marginal cost then the prices would be: (i) $50/MWh for 2800 MW demand, (ii) $50/MWh for 3500 MW demand, and (iii) $80/MWh for 4200 MW demand. Title Page 48 of 54 Go Back Full Screen Close Quit

49 Profit with competitive offers Operating profit per MWh, given competitive offers, for the generators: 2800 MW demand, $50/MWh price baseload ($50/MWh - $20/MWh) = $30/MWh; intermediate ($50/MWh - $50/MWh) = $0/MWh or (not producing) $0/MWh; peaking (not producing) $0/MWh; 3500 MW demand, $50/MWh price baseload ($50/MWh - $20/MWh) = $30/MWh; intermediate ($50/MWh - $50/MWh) = $0/MWh or (not producing) $0/MWh; peaking (not producing) $0/MWh; 4200 MW demand, $80/MWh price baseload ($80/MWh - $20/MWh) = $60/MWh; intermediate ($80/MWh - $50/MWh) = $30/MWh; peaking ($80/MWh - $80/MWh) = $0/MWh or (not producing) $0/MWh. Title Page 49 of 54 Go Back Full Screen Close Quit

50 Profit with competitive offers To find operating profit over day, multiply profit per MWh times production times eight hours and sum across demand levels: 2800 MW demand, $50/MWh price baseload $30/MWh times 250 MW times 8 hours = $60,000; intermediate $0/MWh times production times 8 hours = $0; peaking $0/MWh times 0 MW times 8 hours = $0; 3500 MW demand, $50/MWh price baseload $30/MWh times 250 MW times 8 hours = $60,000; intermediate $0/MWh times production times 8 hours = $0; peaking $0/MWh times 0 MW times 8 hours = $0; 4200 MW demand, $80/MWh price baseload $60/MWh times 250 MW times 8 hours = $120,000; intermediate $30/MWh times 150 MW times 8 hours = $36,000; peaking $0/MWh times production times 8 hours = $0. Title Page 50 of 54 Go Back Full Screen Close Quit

51 Auction rules for offers Submit unit specific offers, one for each of three units i=1,2,3 for each of three pricing intervals t = 1,2,3 in each portfolio. No physical withholding, so maximum quantity must equal the capacity. For each unit i and pricing interval t specify an affine offer with an intercept a it and slope b it : p it (Q it )=a it + b it Q it. Recall that the interpretation of the offer is that if unit i is asked to produce Q it then the price paid will be at least p it (Q it )=a it + b it Q it. The offer must be non-decreasing: that is, b it 0, this means that the offer is the derivative of a convex function. A competitive offer would involve setting the offer equal to the constant marginal cost: set the slope b it = 0 for all i and t, and set the intercept a it equal to the marginal cost of production. Title Page 51 of 54 Go Back Full Screen Close Quit

52 What offers would improve your profits? Your task is to increase profits over day compared to competitive offers. Each group will submit an to with subject EE394V homework group g where g is the group number (1,...,10), with the specification of parameters in the body of the as follows: a 11,b 11,q 11,a 12,b 12,q 12,a 13,b 13,q 13 ; a 21,b 21,q 21,a 22,b 22,q 22,a 23,b 23,q 23 ; a 31,b 31,q 31,a 32,b 32,q 32,a 33,b 33,q 33 ; That is, data is comma delimited, with a semi-colon at the end of each line. Unit 1 is specified on line 1, with the parameters for the three intervals appearing in succession. Physical withholding is prohibited, so q it must match the capacity of the unit you are specifying. You are completely free to specify the a it but b it 0: No market monitor except for capacity! Title Page 52 of 54 Go Back Full Screen Close Quit

53 Timeline must be received by 10pm on Tuesday, February 9. cc the to everyone in the group, so that I know who is in each group. If you are late or if the format of your deviates from the required specification, a competitive offer will be submitted instead. We will discuss in class on Thursday, February 11. Each week, will will follow a similar pattern, with offers due on Tuesday and discussion the following class or in the following week. Title Page 53 of 54 Go Back Full Screen Close Quit

54 Homework exercise: Due Thursday, February 11, at beginning of class Assume that the costs c i and benefits b k are differentiable and that offers and bids are competitive. Assume that the market clearing conditions involve a price P, generator quantities Q i, and demand quantities D k. Show that the market clearing quantities maximize the profits ( producer surplus ) of the generators and maximize the consumer surplus of the loads. That is, show that: (i) for each generator i, for the given price P, the choice Q i maximizes the producer surplus P Q i c i (Q i ) over choices of Q i, and (ii) for each demand k, for the given price P, the choice D k maximizes the consumer surplus b k (D k ) P D k over choices of D k. So, if profits are maximized given the prices P, why would any market participant choose to offer non-competitively? Title Page 54 of 54 Go Back Full Screen Close Quit